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THE COSTS AND BENEFITS OF COVID-19 LOCKDOWNSDr Martin LallyCapital Financial Consultantslallym@xtra.co.nz 16 December 2020AbstractThis paper considers the merits of the New Zealand government adopting lockdowns rather than a milder mitigation policy in response to the covid-19 pandemic, with the actions of individuals and other governments taken as given. Based upon the information available at the time of the first lockdown in March, the cost in the form of GDP losses per Quality Adjusted Life Year (QALY) saved from doing so was well in excess of all generally used values for a QALY, and therefore the lockdown policy was inconsistent with those generally used values. Consideration of alternative parameter values and recognition of factors omitted from the analysis would not likely reverse this conclusion. Taking account of information that has since become available strongly reinforces this conclusion. Applying the same analysis to the Auckland lockdown, it is not possible to clearly conclude either way on the merits of that decision based upon information available both at the time and subsequently. Applying the same analysis to the possibility of future outbreaks that could not be contained with only quarantining of cases, lockdowns might be justified if the proportion of the national GDP generated by the region subject to lockdown were sufficiently small (with Auckland currently being the upper limit) and the probability of conditions giving rise to a second future lockdown is not high. As we move closer to the mass vaccination point in time, this proportion goes to zero.The helpful comments of Glenn Boyle, Amanda Breton, Paul Calcott, Eric Crampton, John Gibson, Dave Heatley, Shaun Hendy, Mark Johnson, Michael Reddell, Jack Robles, and Adrian Slack, and those of participants in a seminar at Victoria University on 18 September, are gratefully acknowledged. Agreement with the conclusions is not implied and should not be inferred. IntroductionAs with most other countries, early this year, the New Zealand government implemented a substantial curtailment of normal activities in order to reduce the death toll from Covid-19 (“lockdown”). Since then, the curtailments have been substantially relaxed, apart from being substantially but temporarily reinstated for the Auckland area. This paper presumes that this lockdown policy will save lives and attempts to estimate the cost of such a policy (especially in terms of the GDP losses) relative to the additional lives saved, so as to compare this to generally employed ratios for guiding public health spending in general. It also conducts this exercise at four different points: a forward-looking exercise at the time of the first lockdown (23 March), a retrospective view of that decision, a retrospective view of the second lockdown decision on 12 August, and a forward-looking view at the present time (30 November). Looking Forwards on 23 March2.1 Expected DeathsAt the time of the first lockdown in New Zealand, death toll forecasts had already been presented by Ferguson et al (2020) for the UK. They estimated that a China style “suppression” policy (which might have to be maintained for several months) would yield about 20,000 deaths in the UK (ibid, Table 4) whilst that from a much milder “mitigation” policy (involving case isolation, quarantining of members of their households, and social distancing for the elderly) would yield about 250,000 deaths (ibid, page 16). Ferguson et al therefore recommended suppression, and their recommendation appears to have significantly influenced UK government policy. However, and remarkably, the paper contains no information on the costs of adopting a suppression policy rather than a mitigation policy. Implicitly, their goal seems to have been to minimize the number of deaths regardless of the cost. This is not a rational goal, it is inconsistent with the small GDP share allocated to health care in every society, and could involve inflicting on a country a ‘cure that was worse than the disease’. Applied to New Zealand, Ferguson et al’s (2020) expected saving in lives from a suppression policy rather than a mitigation policy (of 230,000) would translate into 17,000 lives saved in New Zealand if the results here were proportional to the difference in populations (5m for NZ and 68m for the UK). These 17,000 people would be mostly old and almost all would be suffering from pre-existing conditions. Ferguson et al (2020, page 7) also estimate the casualties at 510,000 from no control measures, implying a worst case saving in lives from a suppression strategy over (totally ineffective) mitigation efforts at 490,000, implying a saving in lives in New Zealand of 36,000. As noted by Ferguson et al (2020, page 7), this presumes no change in people’s behavior in the face of the pandemic, which is extremely unrealistic.Shortly afterwards, and at the commencement of the first lockdown here, predictions for New Zealand were presented by Blakely et al (2020), who estimated the deaths from an “eradication” policy (equivalent to lockdown) at 500 and those from a “mitigation” policy (involving isolation of the over 60s) at 6,500-13,000. The expected saving in lives from eradication (if successful) rather than mitigation was therefore 6,000 - 12,500. Blakely et al (2020) also proffered an estimate for the average residual life expectancy of the victims sans Covid-19 at five years, implying the number of life years saved at 6,000*5 = 30,000 to 12,500*5 = 62,500. They did not reduce this further to reflect the imperfect health of these victims sans Covid-19, so that their estimate of the Quality Adjusted Life Years (QALY) saved per victim was still five years. However the fact that most of the victims had co-morbidities would imply the need for some such deduction, and Miles et al (2020, page 69) subsequently proposed a reduction of 20%. The saving in QALYs would then range from a lower bound of 6,000*5*0.8 = 24,000 to an upper bound of 12,500*5*0.8 = 50,000. Blakely et al’s (2020) worst case scenario was 30,600 deaths, yielding a saving in QALYs of 30,100*5*0.8 = 120,400 from eradication (if successful) rather than mitigation. Blakely et al (2020) recognized that eradication may fail, and assigned a 25% probability to this, so that the expected QALY saving from seeking eradication is [9750 – (.25*9750 + .75*500)]*5*0.8 = 27,750. Interestingly, Blakely et al (2020) recognize that lockdowns involve costs (health expenditures and GDP losses), that these are relevant to the decision, and they estimated the maximum expenditure consistent with their valuation of a QALY at $2.5b.Contemporaneously, James et al (2020a, Table 2) presented predictions of the death tolls in New Zealand from a range of possible control strategies. Lockdown corresponds to the last strategy in their Table 2, and involves predicted deaths of 20 (0.0004% of the population). However, none of the other strategies examined by them in their Table 2 corresponds to “mitigation” as defined by Ferguson et al (2020) and Blakely et al (2020). The fourth strategy in their Table 2 embodies case isolation and quarantining of members of their households, but not also social distancing of the elderly, and therefore (unsurprisingly) yields a predicted death toll much higher than that of Blakely et al (2020), at 62,500. Table 2 also includes a no control strategy, with a predicted death toll of 83,000 (1.67% of the population). James et al (2020a, page 7) do consider a strategy they describe as mitigation, with a predicted death toll of 25,000 (0.508% of the population), but it involves a combination of periods of low control (case isolation plus household quarantining) with periods of lockdown as required to keep the number of cases within the capacity of the hospital system. I therefore treat the work of Blakely et al (2020) as providing the best estimate of the death toll from mitigation (as defined by them) that was available on 23 March. Interestingly, and unlike Blakely et al but like Ferguson et al, James et al do not present any information on, or even recognize the relevance of, costs of adopting the various strategies considered by them, and therefore their goal seems to be to minimize the number of deaths regardless of the cost.2.2 Expected CostsTurning now to the costs of the lockdown policy, this principally takes the form of lost GDP. Shortly before the pandemic arose, in December 2019, The Treasury (2019, page 3) forecasted New Zealand’s real GDP growth rates for 2020-2024 at the rates shown in the first row of Table 1. This is an estimate of growth under the pandemic but with no curtailment of economic activity to reduce its spread. In April 2020, they released figures revised to reflect several different scenarios differing by the time for which Level 3 and 4 operated and the situation elsewhere in the world, of which the least severe (S1), the most severe (S3), and an intermediary scenario (S2) are shown in the other rows of the table (The Treasury, 2020a, page 7). Arbitrarily designating 2019 GDP as 100, the GDP results under these paths are as shown in Table 1. The aggregate difference between the no curtailment scenario and S1 is 539.1 – 511.4 = 27.7, which represents 28% of New Zealand 2019 GDP. Since New Zealand’s 2019 GDP was $311b, this is $87b. Even this figure may be too low because the 2024 GDP level under S1 is below that from the no-curtailment scenario (112.3 versus 113.3) and such GDP losses relative to the counterfactual would continue until these levels were equal. All other scenarios considered by The Treasury yield even more severe GDP losses.Table 1: GDP Forecasts___________________________________________________________________________20202021202220232024Sum___________________________________________________________________________Dec 2019 Forecasts2.2%2.8%2.7%2.5%2.4%Implied GDP102.2105.1107.9110.6113.3539.1April 2020 Forecasts: S1-4.5%-2.5%10.0%5.5%4.0%Implied GDP95.593.1102.4108.1112.3511.4April 2020 Forecasts: S2-8.0%-3.0%13.0%5.5%4.0%Implied GDP92.089.2100.8106.4110.6499.0April 2020 Forecasts: S3-8.0%-23.5%30.0%13.0%6.5%Implied GDP92.070.491.5103.4110.1467.4___________________________________________________________________________Some of these GDP losses of $87b would have arisen without any New Zealand government-imposed restrictions, because some people would have reduced their interactions with others anyway; for example, a foreigner electing not to make a trip to New Zealand that they would otherwise have made, or a New Zealander choosing to avoid cafes. Further losses would have arisen due to the additional actions of foreign governments; for example, foreign governments preventing their citizens from making foreign trips. Further losses would have arisen if the New Zealand government had followed merely a mitigation strategy, which includes border closures. Finally, further losses would have arisen from the New Zealand government instead following a lockdown strategy. It is only the last category of these losses that can be attributed to the New Zealand government choosing a lockdown policy rather than a mitigation policy. Estimating the proportion arising from this last category is problematic. However, the difference between the S1 and S3 scenarios shown in Table 1 is simply in how long Levels 3 and 4 persist (two months in S1 and twelve months in S3, which is a multiple of 6). In addition, going from S1 to S3 adds a further loss in GDP equal to 44% of the 2019 GDP (511.4 – 467.4), and this relates wholly to the last category of loss (the loss from lockdown rather than mitigation by the New Zealand government, with all other effects held constant). This suggests that the GDP loss within S1 arising from the New Zealand government’s adoption of a lockdown rather than a mitigation policy would be about 1/5 of that in moving from S1 to S3, which is 44%/5 = 8.8%. So, under S1, the GDP loss from lockdown rather than mitigation would be 8.8% of 2019 GDP, i.e., $27b. The remainder of the 28% GDP loss in S1 is due to the other contributors listed above. 2.3 Cost per QALY SavedCoupling this $27b loss with the highest estimate of QALYs saved of 120,400, the cost per QALY saved would then be $224,000 as follows: $27b120,400=$224,000 (1) This is a lower bound on the cost per QALY saved from lockdown rather than mitigation because the GDP loss of $27b is a lower bound arising from this policy choice (because it reflects The Treasury’s least severe scenario S1) and the 120,400 QALYs saved is the upper bound from this policy choice (from Blakely et al’s upper bound). Conover (2020) notes that it is conventional practice in the US to further discount QALY figures by about 3% per year to recognise that an additional life year saved immediately is more valuable than one saved later. This would depress the denominator in equation (1), and therefore raise the cost per QALY saved. Furthermore, Pharmac (2015, pp. 51-52) recommends that all costs and benefits in health expenditure assessments be discounted by 3.5% per year, based on the five-year average real government bond rate. The longest term inflation-indexed bonds for which data is available for the last five years are those maturing in September 2035, and the average rate over the last five years has been 1.58%. Since the GDP losses are also spread out over time, the same discounting adjustment should be made there, but the GDP losses are on average much closer to the present moment, and therefore the effect of such discounting for the future adjustments in both the denominator and numerator of equation (1) would slightly raise the cost per QALY saved (by about 2%).Turning now to the value of a QALY saved, Blakely et al (2020) used a QALY value of $45,000, consistent with the BODE3 project (which estimates the cost effectiveness of various health interventions in New Zealand). Kvizhinadze et al (2015, page 3) used the same figure for New Zealand, based upon a WHO recommendation of per capita GDP. However, with a current population of 5.0m and 2019 GDP of $310b, New Zealand’s current per capita GDP is $62,000. Pharmac (2015, page 12) does not prescribe a value, because other factors influence its decisions and any value would depend upon its budget. However, Pharmac’s highest average cost per QALY saved in recent years is $27,027 in 2016/17, which implies $33,306 in 2020. In evaluating the merits of extending the Level 4 lockdown for five extra days, Heatley (2020, page 9) used the figure just referred to. In the interests of being conservative, I use the highest number here of $62,000. A related concept is the “Value of a Statistical Life” (VOSL), which values all lost years of an average aged person’s life. This is generally employed in decisions on road safety and the current value is $4.37m (Ministry of Transport, 2019, page iii). To convert to a value per year (called the Value of a Statistical Life Year or VOSLY), which can be compared to the value of a QALY, it would be necessary to first determine the average residual life span of a road accident victim. The median age of a victim is 41, and 2/3 are male. The median residual life span of a 41 year old is 43 years for a female and 40 for a male. This implies an average residual life span for a median aged road victim (of age 41) of 41 years. Applying Pharmac’s recommended discount rate of 1.58% (see penultimate paragraph), the value per QALY would be V such that$4.37m=V1.0158+…+V(1.0158)41The solution is V = $146,000. This $4.37m estimate of the VOSL was derived from a 1991 survey (Ministry of Transport, 2019, page 14), as is usually the case for such estimates, whilst the value of a QALY comes from public health experts or deduced from the health expenditure decisions by government; the latter would then seem to be more relevant to the present situation involving a health policy choice by government. Furthermore, amongst the authors I am aware of who consider both approaches (Miles et al, 2020; Gros, 2020), both favour the latter approach over the VOSL approach, with Gros (2020, page 6) noting that it the approach that is “practiced routinely by the medical profession” and Miles at al (2020, page 76) that it is consistent with the approach to other health expenditures. Thus I favour a QALY value of $62,000 but also consider the effect of using the VOSL derived figure of $146,000.Interestingly, some analyses of the Covid-19 issue have been conducted by simply multiplying the VOSL by the expected saving in lives (for example, Chapple, 2020; Thunstrom, 2020; Holden and Preston, 2020). This may reflect the mistaken view that Covid-19 victims have a typical average residual life span and perfect health sans Covid-19, which is not the case, and would therefore significantly overestimate the benefits in QALYs saved. Alternatively, it may reflect the view that all lives saved are equally valuable, which implies that one would spend as much to extend the life of a person by one day as one would spend to extend the life of a different person for ten years. If the latter interpretation is correct, it is inconsistent with prevailing views in the health sector. In particular, Pharmac (2015, page 36) recommends that residual life expectancies be estimated and further adjusted for the patient’s quality of life, and this is also the orthodox view amongst New Zealand public health experts (see Kvizhinadze et al, 2015; Blakely et al, 2020). Since the lower bound on the cost per QALY saved from lockdown rather than mitigation is $224,000 as shown in equation (3), use of a QALY value of $62,000 implies that the cost of adopting a lockdown policy rather than a mitigation policy per additional QALY saved would be at least 3 times the QALY value of $62,000. Had one instead used Blakely et al’s (2020) expected extra lives lost of 7,000, and therefore 7,000*5*0.8 = 28,000 QALYs saved, the cost per QALY saved would have risen to $960,000, which is 15 times the QALY value of $62,000. Had one also used The Treasury’s S2 scenario rather than its S1 scenario (see Table 1), which could be viewed as a mid-case scenario, the extra GDP loss would have been 12.4% of the 2019 GDP entirely due to lockdown rather than mitigation by the New Zealand government (511.4 – 499.0), which would imply a total GDP loss from lockdown over mitigation of 21.2% (8.8% in S1 plus 12.4% extra in S2), which would be $66b. The cost per QALY saved would then be $2.36m as follows: $66b28,000=$2.36m (2) This is 38 times the QALY value of $62,000. Table 2 shows a range of combinations of incremental deaths and incremental GDP losses. For S3, the total GDP loss from lockdown over mitigation is 52.8% of 2019 GDP (8.8% in S1 plus 12.4% extra in S2 plus 31.6% extra in S3), which would be $164b. In Table 2, all of these costs per QALY saved are a multiple of the QALY value of $62,000. Furthermore, all of them are in excess of the QALY value of $146,000 derived from the VOSL. In relation to the central estimate, involving the saving of 9,000 extra lives at an estimated cost of $62b, consistency in government action would require spending $62b to extend the lives of 9,000 people suffering from heart disease, cancer or diabetes, which is three times the entire annual government spending on health care in New Zealand. Table 2: Cost per QALY Saved by Lockdown Versus Mitigation___________________________________________________________________________Incremental Deaths6,0007,00030,100_______________________________________________________________________GDP Loss (S1): $27b$1.12m$0.96m$0.22mGDP Loss (S2): $66b$2.75m$2.36m$0.55mGDP Loss (S3): $164b$6.83m$5.86m$1.36m___________________________________________________________________________It is implicit in all of this that the worst case scenario is the 30,100 additional lives in Blakely et al (2020), and that the value of a QALY does not exceed $62,000. However, if one focused on a worst-case scenario of 47,000 additional lives saved from lockdown versus mitigation, and a GDP loss from lockdown over mitigation of as little as $27b as above, the cost per QALY saved would be $144,000 as follows: $27b47,000(5)(0.8)=$144,000Additionally using a value per QALY of $146,000 from the VOSLY approach, the decision to lockdown would have then (just) been justified within this framework. However this would require the lowest estimate for the GDP loss, the highest possible value for a QALY, and a loss of lives well beyond Blakely et al’s (2020) worst case scenario.Returning to the GDP loss, whatever the figure is, it may not be ‘evenly’ shared (through the tax system in the usual way with medical expenditures to extend lives).? It may fall largely on two groups. The first own businesses and would suffer a loss in profits, possibly to the point the business fails. The second would lose their jobs and remain unemployed for some period. By adopting a lockdown rather than a mitigation policy, in order to moderately extend the lives of 9,000 largely old and sick people, these two other groups might suffer substantial financial losses. The extent to which this actually occurs will depend upon the extent to which their fellow citizens compensate them for these losses. Furthermore, even if the government does intervene in an attempt to protect some vulnerable groups, other vulnerable groups may be inadvertently hurt. For example, if rent payments were suspended by government for some period, landlords will suffer and some of them will be retired people who derive a considerable proportion of their modest incomes from this source. In addition, prospective landlords will not rent out their properties, to the disadvantage of both parties.This analysis omits many relevant features, some of which would reduce the cost per QALY saved by a lockdown rather than a mitigation policy. Firstly, with a mitigation rather than a lockdown policy, the health system may be so overwhelmed by Covid-19 cases that many other people may die or suffer reduced quality of life from lack of care. Secondly, there are direct medical costs of dealing with the demand surge from a mitigation rather than a lockdown policy. Boyd (2020, Table 3) estimates these at $1b if the system is saturated with cases, which only trivially lowers the cost per QALY saved. Thirdly, lockdown reduces road deaths and pollution. Layard et al (2020, page 2) examines this issue and concludes that these additional factors are slight relative to the higher deaths from Covid-19.This analysis also omits many features that would raise the cost per QALY saved. Firstly, some of the lives seemingly saved from a lockdown rather than a mitigation policy would be lost anyway if the lockdown restrictions were lifted before a vaccine arrived and then gave rise to additional deaths (Holden and Preston, 2020). So, the additional lives saved by lockdown rather than mitigation would be overestimated. Secondly, lockdown increases unemployment and therefore leads to an increase in a host of attendant problems: addiction, crime, domestic violence, mental health problems, and premature death. Thirdly, the normal operation of the health care system would be disrupted by a lockdown policy, leading to deaths that would not otherwise occur. In respect of the UK, Miles et al (2020, page 68) notes that referrals for cancer investigations were 70% down in April 2020, and outpatient visits were 64% down between mid March and early June. Some of these reductions are a consequence of the pandemic rather than pursuit of a lockdown policy by government, but at least some of them will have arisen from the policy choice. Fourthly, there are adverse long-term consequences for the Covid-19 student cohort from not attending school. Fifthly, lockdowns deprive people of liberties that they would otherwise exercise. All of these raise the cost per QALY saved. In addition, the appropriate cost from the GDP losses is not the GDP losses themselves but the “economic surplus”, which is the consumer surpluses (the aggregate amount consumers are willing to pay in excess of the amounts actually paid for all goods and services) plus the producer surpluses (the aggregate amounts received by producers less the amounts they would be willing to accept to produce at this level), and this sum could be more or less than the GDP losses. Again, there is considerable room for large adjustments before lockdown involves a cost per QALY saved no higher than generally employed value of a QALY. All of this suggests that, based upon information available to the government at the time of the first lockdown in March, the cost per QALY saved of a lockdown rather than a mitigation approach by the New Zealand government would be dramatically inconsistent with long-established views on the value of a QALY. Consideration of non-quantifiable factors does not appear to alter this conclusion. There are ethical judgements here, reflected in the conventional QALY approach to health decisions. In particular, a life with one year left is worth ten times that of a life with ten years left. Naturally, there is room for different views, but one could not reasonably view all lives equally because that would lead to being willing to spend as much to extend the life of a person by one day as one would for a different person for ten years. I limit myself here to applying the QALY methodology, which is used by Pharmac (2015) and public health experts (Kvizhinadze et al, 2015; Boyd et al, 2018; Blakely et al, 2020).2.4 Real Options AnalysisThis analysis characterizes the policy choice as lockdown or mitigation in March 2020, without the option of subsequently changing actions, i.e., initial lockdown could have been subsequently displaced by mitigation if the deaths (had mitigation been initially undertaken) were much lower than expected, and initial mitigation could have been subsequently replaced by lockdown if the deaths proved to be much larger than expected. Interestingly, in acknowledging that it did not apply quantitative cost-benefit analysis to assist in its lockdown decision, the National Crisis Management Centre (2020, para 8) justifies this on the grounds that it “may not fully capture the dynamic nature of the information and choices available at particular points in time..”I therefore modify the previous analysis, to allow for this possibility of changing the decision as information about the death toll unfolds. Blakely et al (2020) provide a range of possible deaths from mitigation of 6,500 to 30,600 along with deaths of 500 if lockdown succeeded. I adopt these numbers. I also assume that the initial decision of 23 March is reconsidered after one month, at which point the death toll outcome from mitigation throughout is then predictable, that switching from lockdown to mitigation at this point halves the GDP losses from the lockdown ($27b), that doing so does not lessen the death toll incurred from adopting mitigation, that switching from mitigation to lockdown at this point does not reduce the GDP losses from the lockdown, and that doing so halves the death toll. The analysis appears in Table 3 below, with the first column showing the initial decision, the second column the deaths if mitigation were pursued throughout, the third column the decision after one month, the fourth column the cost from doing so (deaths scaled up by four to QALYS and multiplied by the value per QALY of $62,000, plus GDP losses relative to mitigation throughout), and the fifth column the optimal choice in one month. As shown in the first section of the table, if lockdown is initially adopted, the optimal choice in one month is always to mitigate, whether deaths from mitigating throughout are low or high. The costs here range from $15.1b if deaths are low to $21.1b if deaths are high. Similarly, as shown in the second section of the table, if mitigation is initially adopted, the optimal choice in one month is to continue doing so, whether deaths from mitigating throughout are low or high. The costs here range from $1.6b if deaths are low to $7.6b if deaths are high. So, the optimal decision now is to mitigate, because its outcomes ($1.6b to $7.6b) are superior to their counterparts under lockdown now ($15.1b to $21.1b). Furthermore, mitigation now is the optimal choice under uncertainty (unknown probabilities) as well as risk (probabilities known). In addition, not only is mitigation initially the optimal choice but so too is it at the reassessment point in one month.Table 3: Optimal Decisions___________________________________________________________________________Initial DecisionDeathsNew DecisionCostChoice___________________________________________________________________________6,500Lockdown500*4*$62,000 + $27b = $27.1b6,500Mitigate6,500*4*$62,000 + $13.5b = $15.1bMitigateLockdown30,600Lockdown500*4*$62,000 + $27b = $27.1b30,600Mitigate30,600*4*$62,000 + $13.5b = $21.1bMitigate___________________________________________________________________________6,500Lockdown3,500*4*$62,000 + $27b = $27.9b6,500Mitigate6,500*4*$62,000 + 0 = $1.6bMitigateMitigate30,600Lockdown15,500*4*$62,000 + $27b = $30.8b30,600Mitigate30,600*4*$62,000 + 0 = $7.6bMitigate___________________________________________________________________________The only situation in which lockdown is even optimal in one month is when the worst case scenario is sufficiently far above that presented by Blakely et al (2020). This requires deaths from mitigation throughout of 54,000, which induces continuation of lockdown after one month in the highest death toll scenario (initial lockdown plus deaths under mitigation throughout of 30,600). Even so, the optimal decision now is still to mitigate, and to maintain that policy at the reassessment point. Even a maximum death toll of 108,000 under mitigation throughout would still leave mitigation now superior because its costs would be lower in all death rate scenarios except the highest and no worse in the highest. Thus, there is no remotely plausible probability distribution for deaths under mitigation that would justify lockdown now. The analysis assumes that switching from mitigation to lockdown after one month halves the death toll, but this is inconsistent with the analysis in the Appendix that failing to lockdown quickly implies that doing so later is pointless. In this case, the deaths from this situation would be the same as those from never locking down, this makes the switch even less favourable, and this does not alter the decision at this point because mitigation at this point is already favoured. Blakely et al (2020) also alludes to the possibility of lockdown failing, but consideration of this possibility makes lockdown even less favourable and therefore does not change the result here: mitigating initially was optimal, and remained so even if the decision could be changed as death rate information unfolded, because the worst case death toll scenario was insufficiently high.This analysis shows that, contrary to the claims of the National Crisis Management Centre, quantitative cost-benefit analysis readily allows for the dynamic features of the lockdown decision process and that doing so affirms the conclusion that lockdown was not warranted based on the information available on 23 March 2020.Looking Back to 23 March3.1 DeathsIn the months since the first lockdown, significant new information has become available, on deaths, GDP losses, and other issues. In respect of deaths under its lockdown policy, New Zealand’s toll to date has been 25. This is vastly less than Blakely et al’s (2020) estimate of 500 from an eradication policy. In respect of the deaths that would have occurred under a mitigation policy, the Blakely et al (2020) forecasts are displaced by estimates from the actual experience of other countries. One such source of evidence is the variation in death rates between US counties that did and did not lockdown; Gibson (2020a) finds no statistically significant difference, after controlling for various other factors. Another source of evidence is the timing of the death peaks relative to the timing of the lockdowns, in various countries or regions that implemented lockdowns; Stone (2020) examines Spain, France and Lombardy and finds in all three cases that these intervals are shorter than the minimum 20 day period between infection and death, which implies that the lockdowns did not induce the reduction in the death rate and therefore at most enhanced the rate of decline. Similarly, Homburg (2020) uses death data to estimate the peak of the curve for nine countries, of which seven adopted lockdowns. For each country, he deducts 23 days from the death peak to determine the infection peak (because the interval from infection to death is about 23 days), and then compares this infection peak to the timing of the lockdowns. In all countries which adopted lockdowns, the lockdown occurred after the infection peak, with a median difference of four days (ibid, Table 2). Again, this reveals that the lockdowns did not cause the decline in the deaths and therefore at most enhanced the rate of decline. Gibson (2020b, pp. 3-5) extends this analysis to the 34 OECD countries and uses a longer time period to estimate the peak infection date, and finds that this peak predates the maximum stringency of their restrictions in half of these countries.A more direct source of evidence is variation in death rates across countries that did and did not lockdown. Chaudhry et al (2020) examine the 50 countries with the highest case counts as at 1 April, and regresses cross-country death rates per 1m of population (up to 1 May) on a number of independent variables, including various measures of government intervention, and find that none of these latter variables were statistically significant. Gibson (2020b) conducts a similar analysis, using the 34 OECD member countries, death rates up to 18 August, and various independent variables including the average level of government restrictions over the period of the crisis. He finds that policy stringency (averaged over the whole crisis period) is not statistically significant in explaining cross-country variation in death rates (ibid, Table 2). He also examines average stringency both before and after the estimated infection peak for each country, and finds mild statistical significance for average stringency prior to the estimated infection peak along with a negative coefficient (ibid, Table 2). However, using that estimated coefficient, the estimated saving in lives lost from New Zealand instead adopting restrictions at the Swedish level would have been only 310 (ibid, page 6). He also uses average stringency in other countries within the same OECD group as an instrumental variable, to test for reverse causality between stringency and death rates, and finds no evidence of reverse causality.Appendix 1 further examines this issue using data to 30 November and concludes that the expected increase in New Zealand’s death rate per 1m from moving from its lockdown policy to a mitigation policy like that of the least stringent European country (Iceland) is no more than 317N - 5, where 317 is the upper bound on the 95% CI from the New Zealand death rate under a mitigation policy up to 30 November, N is the ratio of the cross-country average death rate for European countries at the termination of this pandemic to the current (30 November) average, and 5 is the New Zealand death rate to date (30 November) from its lockdown policy. To a close approximation, this is 312N. Since New Zealand has 5m people, the expected deaths avoided by lockdown are up to 1,560N.These estimates presume that Covid-19 deaths are accurately recorded. However, some deaths may be mistakenly attributed to another cause, or mistakenly attributed to Covid-19, with the latter error possible simply because most victims have co-morbidities. By analogy, if a person is shot in the heart and then the head, and then dies, the cause of death may not be the head shot. In addition, lockdown disrupts the normal operation of the health system, leading to some deaths in lockdown countries that would not otherwise have occurred, and these should be included in the incremental deaths from lockdown. In addition, mitigation increased the load on hospitals, leading to more deaths from other causes in mitigation countries, and these should be included in the incremental deaths from mitigation. An estimate of the Covid-19 deaths that accounts for all of these phenomena is the actual deaths in 2020 less the predicted number sans Covid-19 (“excess deaths”). The Euromomo Network has done so and estimated the number of deaths across 18 European countries progressively through 2020, 2019 and 2018 relative to a prediction (“baseline”). The excess deaths to 15 September relative to the baseline was 210,000 for 2020, 65,000 for 2019 and 120,000 for 2018. By contrast, the deaths attributed to Covid-19 across these 18 countries (to 15 September) were 169,000. Thus, if the baseline were used, the excess deaths in 2020 would be 210,000 and therefore the deaths attributed to Covid-19 of 169,000 would be too low by 20%. However, the baseline is an imperfect prediction, as evidenced by the results for 2018 and 2019 (which would be zero if the predictions were accurate). This is simply a consequence of the fact that deaths in a typical year are about 3m, so that the prediction error of 120,000 for 2018 is a small proportion. Furthermore, even if the baseline were used, the fact that the first deaths attributed to Covid-19 in these countries did not occur until 15 February suggests that excess deaths be counted from only that point, which would yield excess deaths from 15 February to 15 September of 190,000. This is sufficiently close to the actual deaths attributed to Covid-19 to accept the latter figure. 3.2 QALYsThe estimate of avoided deaths is now converted to an estimate of expected QALYs saved. In respect of the average residual life expectancy of these victims sans covid, I focus upon the European countries used to estimate the additional death toll under mitigation rather than lockdown. I start with Sweden. The age distribution of the covid-19 victims is shown in the first two columns of Table 4, and the residual life expectancy (RLE) of Swedish people in each such age group is shown in the third column. Using this data, the average residual life expectancy of the Swedish victims sans covid-19 is 10.9 years. This calculation assumes that the victims are typical of Swedish people of the same ages. However, they differ in two very significant features.The first of these features is that a large proportion of the victims were residents of nursing homes, whose average residual life expectancy sans covid-19 was very low and might be even lower than suggested by their ages. If so, conditioning on residency of a nursing home as well as age would reduce the average residual life expectancy of the victims. In respect of Sweden, Stern and Klein (2020, page 5) estimate that 53% of the covid-19 victims aged at least 70 were residents of nursing homes, and that their average residual life span sans covid-19 was only seven months (ibid, pp. 16-17). Conservatively treating this subset of victims as the oldest in Table 4, they represent the entire 85+ group (47%) plus additional victims in the 80-84 group constituting 6% of the entire set of victims (6/21 of that group). Replacing the residual life expectancy of these people by seven months (0.6 years), the average residual life expectancy calculated from the data in Table 4 would fall to 7.7 years as shown in the fourth column of Table 4. By contrast, if this nursing home group were spread through the 70+ groups in proportion to the size of these groups, 28% would be in the 85+ group, 13% in the 80-84 group, and 13% in the 70-79 group. Replacing the residual life expectancy of these people by 0.6 years, the average residual life expectancy calculated from the data in Table 4 would fall to 6.4 years. The second unusual feature of these covid victims is that they were unusually unwell, even for their age; as noted in footnote 1, 98% of the victims had at least one co-morbidity, which is presumably well in excess of the rate for the general population of the same age distribution. A common such ailment was type 2 diabetes. The NHS (2018, Figure 8) provides estimates for the increase in mortality risk from this disease (relative to the general population) by age and sex. Averaging over these categories, the increase is about 50%. However the group of interest here excludes those in nursing homes, because the estimate for the residual life expectancy of these victims already reflects co-morbidities. This exclusion lowers the average age of the remaining victims, and suggests an increase in their mortality risk of about 80%. In addition, a person with a residual life expectancy of 10 years (the average for the covid-19 victims) would have a current mortality risk of about 5% over the next year, growing at about 11% per year compounded: RLE=.051+1-0.05[.051.1](2+1-.05[.051.11.1](3)+…=9.52 yrsRaising this initial mortality risk by 80%, from 5% to 9%, along with the same growth rate of 11%, reduces the residual life expectancy from 9.52 yrs to 6.68 yrs, i.e., a reduction of 30%. A similar percentage reduction applies to the average residual life expectancy of a group. As noted above, virtually all of the victims had at least one co-morbidity, and multiple co-morbidities would reduce the average residual life expectancy of a group by even more than estimated here. Allowing for this additional feature of the covid-19 victims is simplified by the fact that virtually all covid-19 victims had co-morbidities. So, the subset of Swedish victims from nursing homes have their residual life expectancy set as before at 0.6 years (which will also reflect their co-morbidities), and all others have their residual life expectancy reduced by (conservatively) 30%. The results are shown in the penultimate column of Table 4, with the nursing home group conservatively assumed (as before) to be the oldest, and the last column of Table 4, in which the nursing home group is spread throughout the 70+ groups in proportion to their sizes. The average residual life spans are 5.5 and 4.7 years respectively, and a good estimate would lie between these figures. So, starting with 10.9 years, the reduction is to 6.4 - 7.7 years to account for the nursing home group, and then to 4.7 – 5.5 years to additionally account for co-morbidities in the rest.Table 4: Residual Life Expectancy of Swedish Covid-19 Victims___________________________________________________________________________Age Group Victims RLE RLE RLE RLE___________________________________________________________________________0-9279.5579.5579.55(0.7)79.55(0.7)20-291060.2560.2560.25(0.7)60.35(0.7)30-391950.550.550.5(0.7)50.5(0.7)40-494540.840.840.8(0.7)40.8(0.7)50-5916431.331.331.3(0.7)31.3(0.7)60-6940622.422.422.4(0.7)22.4(0.7)70-791268 (22%)14.314.314.3(0.7)4.4580-841219 (21%)9.16.674.722.8085+2747 (47%)6.30.60.62.14___________________________________________________________________________Average (yrs)10.97.75.54.7___________________________________________________________________________This process could be repeated for other European countries. Miles et al (2020, Table 1) performs an analysis akin to that in the first three columns of Table 4, but matching on both age and sex, and thereby estimates the average residual life expectancy of people matching the UK’s covid-19 victims by age and sex at 10.1 years. In addition, Bell et al (2020, Table 3) notes that 40% of UK covid-19 victims were residents of nursing homes. Forder and Fernandez (2011, page 3) estimates the average length of stay for UK nursing home residents at 801 days, and therefore the residual life expectancy of a randomly selected resident would be 400 days (1.1 years). Conservatively treating this 40% subset of victims as the oldest in Miles et al’s set of victims, they correspond to all of those at least 85 in Miles et al’s Table 1. Alternatively, they could be spread across all 75+ groups in Miles et al’s Table 1, in proportion to the size of these groups, and would therefore constitute 54% of each such group. Either way, their life expectancy is set at 1.1 years and all others have their life expectancy reduced by 30%. The result is an average residual life expectancy of 5.7 to 6.3 years. In summary, the average residual life span of covid-19 victims in Europe would be about 10 years if they were typical of their age. However they are not typical, in that co-morbidites were almost universal and the proportion who were residents of nursing homes very high. Allowance for both factors suggests a reduction in average residual life expectancy to about five years. This matches the estimate of Blakely et al (2020). In respect of a discount to reflect the imperfect health of virtually all of these victims sans Covid-19, Miles et al (2020, page 69) use 20% based on prevailing discounts for type 2 diabetes with and without additional problems. In particular, they cite Beaudet et al (2014, Table 3), who favour a quality of life discount of 21% for Type 2 diabetes without complications, and substantial additional discounts for further problems including 9% for heart disease and 16% for stroke. These discounts in Beaudet et al suggest that Miles et al’s 20% discount for an average covid-19 victim is low. Furthermore, a large proportion of the victims were residents of nursing homes, for which the quality of life discount could reasonably be even higher.In conclusion, the QALYs saved by the New Zealand government pursuing lockdown rather than mitigation would then be up to 1,560N*5*0.8 = 6,240N. 3.3 GDP LossesTurning now to the GDP losses, since March 2020, The Treasury has produced two updates to its GDP forecasts. In May, The Treasury (2020b, page 3) revised its December 2019 forecasts and these are shown in Table 5 (along with The Treasury’s December 2019 forecasts). These May forecasts are in aggregate very similar to The Treasury’s S1 forecasts shown in Table 1, which were at that time The Treasury’s least severe scenario. Thus, in May, The Treasury essentially reached the view that its S1 scenario was then the best forecast, implying a GDP loss from the pandemic of $87b, of which the loss due to lockdown over mitigation was 8.8% of GDP or $27b. This represents 31% of the total expected GDP loss of $87b under S1. In September, The Treasury (2020c, page 82) further revised its forecasts as shown in Table 5. For the years 2020-2024, the aggregate GDP shortfall from the pandemic is now 539.1 – 505 = 34.1, which is 34.1% of 2019 GDP. This is larger than the shortfall estimated in May 2020. Furthermore this is not the full extent of the shortfall; this is the accumulation of the annual GDP shortfalls for the September 2020 forecasts relative to the December 2019 forecasts until the annual GDP under the former reaches the latter and this is not even close to equality in 2024 with the implied 2024 GDP from these September 2020 and December 2019 forecasts of 108.0 and 113.3 respectively. Table 5 shows The Treasury’s September 2020 forecasts for 2025-2027 (The Treasury, 2020c, page 82), and extrapolation of the December 2019 forecasts for 2025-2027. Accumulating the GDP forecasts out to 2027 yields 895.6 for the December 2019 forecasts and 847.8 for the September 2020 forecasts, as shown in the table. The difference is 47.8, i.e., 47.8% of 2019 GDP, which is $148b. Even this is not the full extent of the shortfall because the forecast GDP in 2027 under the September 2020 forecasts is still short of that under the December 2019 (extrapolated) forecasts. Table 5: GDP Forecasts___________________________________________________________________________20202021202220232024202520262027___________________________________________________________________________Dec 2019 Forecasts2.2%2.8%2.7%2.5%2.4%2.4%2.4%2.4%Implied GDP102.2105.1107.9110.6113.3116.0118.8121.7Sum539.1895.6May 2020 Forecasts-4.6%-1.0%8.6%4.6%3.6%Implied GDP95.494.4102.6107.3111.1Sum510.8Sept 2019 Forecasts-3.1%-0.5%3.6%3.9%4.1%2.8%2.9%2.9%Implied GDP96.996.499.9103.8108.0111.0114.2117.6Sum505.0847.8___________________________________________________________________________Since the expected GDP loss from the pandemic is now at least $148b rather than $87b, one might suspect that the GDP loss due to lockdown rather than mitigation would be more than the $27b estimated earlier. Two possibilities consistent with this exist. The first is that the expected GDP loss per additional month of lockdown is now larger than the 4.4% of GDP estimated from The Treasury’s March 2020 scenarios. The other possibility is that The Treasury now expects the lockdowns to be longer than the two months envisaged in scenario 1 in March 2020. The lockdowns envisaged by The Treasury in scenario 1 were one month nationally in Level 3 and one in Level 4. Since Level 4 is 60% more stringent than Level 3 (The Treasury, 2020a, page 7), this is equivalent to 2.6 months in Level 3. By contrast the lockdowns to date have been 33 days nationally in Level 4, 18 days nationally in Level 3, and 18 days for the Auckland region in Level 3. Since Auckland constitutes 38% of national GDP, this is equivalent to 77.6 days at Level 3 nationally, which is 2.6 months. So, lockdowns to date match The Treasury’s March 2020 scenario 1. However the fact that the Auckland lockdown happened raises the possibility of further such lockdowns, and therefore that The Treasury’s less favourable GDP forecasts in September compared to March reflect its belief that further lockdowns will happen. Both interpretations are possible. However, in the interests of being conservative, I assume that The Treasury still believes that each month of lockdown rather than mitigation cost 4.4% of GDP, in which case the less favourable forecasts in September relative to May are attributable to greater pandemic costs in the other categories. The expected cost of the lockdown in then at least $27b. At the lower bound of $27b, this represents 18% of the pandemic cost of $148b.Further evidence on this matter comes from natural experiments involving adjoining geographical regions, with only one subject to lockdown. Andersen et al (2020) examine the drop in consumer spending in the early stages of the pandemic (11 March to 5 April), relative to 2 January till 15 February) in both Denmark (which adopted a lockdown policy) and Sweden (which adopted a mitigation policy). They find that the drop in Sweden was 86% of that in Denmark (25% drop versus 29% drop), implying that 14% of the drop in Denmark was due to a lockdown rather than a mitigation policy. In a similar study (Goolsbee and Syverson, 2020) examine adjoining US counties with one area subject to lockdown and the other not; the drop in consumer activity in the latter area was 88% of the former, implying that only 12% was due to the lockdown. This evidence is compatible with the conclusion that the lockdown versus mitigation decision constituted at least 18% of the pandemic cost.3.4 Cost per QALY SavedIn summary, the QALYs saved by locking down rather than mitigating are estimated at up to 6,240N whilst the associated GDP losses are expected to be at least $27b. If these GDP losses were the only cost of lockdowns, the cost per QALY saved would then be as follows: C= $27b6,240N (3) This is a lower bound on the cost per QALY saved because the QALYs saved of 6,240N are an upper bound and the expected GDP cost is a lower bound. I now attempt to quantify all of the additional costs of mitigation, for which considerably better information has become available since March 25. Firstly, there are the medical costs of those requiring short-term hospitalization under a mitigation policy. Gros (2020, section 2.2) assumes that the entire population of a country becomes infected, and estimates that 20% would require general hospital care at a cost per patient equal to 30% of GDP per capita, and ? of these would also require Intensive Care treatment at a further cost per patient of 60% of per capita GDP, yielding a total cost equal to 9% of GDP. However the assumption that everyone in a population would become infected is excessive. Blakely et al (2020) estimate that the infection rate would not exceed 60% because the epidemic would by then peter out through herd immunity, Boyd (2020, page 3) adopts a base case of 40% based upon the experience from past pandemics, and Aguas et al (2020) estimate it to be even lower. Furthermore, even if Gros’s estimates of the proportion infected requiring medical care were correct, this would (even at a 40% infection rate) imply 400,000 New Zealanders requiring hospitalization (8% of its population), and 100,000 of these requiring an ICU. However, there would not be enough hospital beds or ICUs in the country or medical staff to cater for even a fraction of them, as there are only about 13,000 hospital beds in the country and 440 ICU units (Wilson, 2020). By contrast, Wilson (2020) estimates hospitalized cases at 146,000 and ICU cases as 36,600 under a “worst-case scenario” of 27,600 dead. Since my upper bound on the number of dead under a mitigation policy is 1,560N, this implies 8,000N hospitalised cases and 2,000N ICU cases. I assume (again to be conservative) that the New Zealand hospital system could accommodate all of them. Furthermore, whatever number of cases were accommodated, they would be to a large degree permanently displacing other types of patients so the incremental costs would be even less. Even without displacement, most of these costs would be fixed (staff, buildings, and equipment) and therefore irrelevant. In the interests of being conservative, I assume no displacement of other types of patients and all costs being variable. With New Zealand’s GDP per capita of $62,000, the upper bound on the resulting incremental costs would be $0.2bN as follows:62,0008,000N(0.3)+2,000N0.6=$0.2bNUsing this to modify equation (3), the cost per QALY is now as follows: C=$27b-N$0.2b6,240N (4) Secondly, mitigation gives rise to some survivors who may experience significant long-term adverse consequences. Arnold et al (2020) report that, amongst Covid-19 cases in the UK who were hospitalized, 26% died and 74% of the rest had ongoing problems after 12 weeks, implying a ratio of slow recovery patients to dead of 74*0.74/26 = 2.1. However, this ratio will be too low because it excludes slow recovery patients who were never hospitalized. Using data from the Covid Symptom Study, Couzin-Frankel (2020) estimates that 10-15% of all of those infected do not recover quickly. More recently, and using the same data source, Greenhalgh and Knight (2020) estimate that 10% of those who have tested positive remain unwell after three weeks and a smaller (but unquantified) proportion for months. As of 15 October 2020, there have been 38.6m recorded cases and 1.1m deaths worldwide. So, the ratio of these slow recovery cases to deaths is 38.6m*0.1/1.1m = 3.5. These sufferers seem to be similar to the fatalities in their age and the presence of pre-existing conditions. However, given the lack of precision about this, I conservatively assume a residual life expectancy for these long-term sufferers to be double that of the fatalities, which is therefore 10 years. Furthermore, consistent with the figure used earlier for people suffering from serious pre-existing conditions, their quality of life is thereby initially reduced by 20%. Since Greenhalgh and Knight (2020) imply that some of them do recover, I assume an average loss in their quality of life over their remaining life equal to half of this, which is 10%. Allowing for all this is then equivalent to increasing the QALYs saved from lockdown rather than mitigation by a factor of approximately 2.0 as follows:1(5)(0.8)+3.5(10)(0.1)1(5)(0.8)=4.0+3.54.0=1.88This raises the denominator in equation (4) by a factor of approximately 2.0. There will also be medical costs associated with these long-term sufferers. For example, if each such person’s medical costs average $5,000 per year of their residual life, the cost for 10,000 survivors (5 times the estimated upper bound Covid-19 deaths of 1,560 + 25 = 1,585) for 10 years each is $0.4b. So, making these adjustments to equation (4) reduces the cost per QALY as follows:C=$27b-N$0.2b-N$0.4b6,240(2)NThirdly, mitigation gives rise to work absences amongst those who are infected and must self-isolate. Gros (2020, section 2.1) assumes all members of a population are infected, 30% require a work absence of four weeks and a further 20% require six weeks, leading to a GDP loss of 5% of one year’s GDP:.30452+.20652=.05However the assumption that everyone in a population would become infected is excessive. As noted previously, Blakely et al (2020) estimate that the infection rate would not exceed 60% because the epidemic would by then peter out through herd immunity, Boyd (2020, page 3) adopts a base case of 40% based upon past pandemics, and Aguas et al (2020) estimates an even lower rate. Furthermore, even if Gros’s estimate of 50% of those infected requiring a work absence were correct, this does not yield a proportionate decline in GDP for various reasons. In particular, many of the people required to isolate could still perform their work from home. Furthermore, even where those isolated could not thereby perform their tasks for this period of weeks, other employees of the organization would increase their productivity or hours of work to at least partly compensate, and/or customers of the businesses would experience longer wait times with no loss of output, and/or the absent employees would be able to perform at least some of the work upon their return in addition to their normal workloads. Accordingly, a more reasonable estimate of the GDP loss than Gros’s would involve 40% of the population being infected, and 30% of those requiring isolation able to still perform their jobs at home, and 75% of the rest having their work performed by others or by them upon their return to work or addressed through longer customer queues. The resulting GDP loss would then be only 5%*0.4*0.7*0.25 = 0.35% rather than 5% of one year’s GDP. In dollar terms this is $310b*0.0035 = $1.1b. Modifying the penultimate equation, the cost per QALY saved would be at least as follows: C=$27b-$1.1b-$0.6bN6,240(2)N (5)Whether this exceeds the current QALY value of $62,000 depends upon the value of N. At N = 2, the cost per QALY saved would be $0.99m, which is vastly in excess of the currently employed value. In fact, N would have to be 19 before the cost per QALY in equation (5) fell to $62,000, or 11 to reach a cost per QALY of $146,000, i.e., the eventual death toll in Europe from the pandemic would have to be 19 (or 11) times the death toll to date (30 November). Even the lower multiple of 11 seems very unlikely. To gain some sense about the plausibility of such a value of N, suppose that mass vaccination is one year away (from 30 November 2020). So, for N = 11 to prevail, the expected loss in lives in Europe over the next year would have to be 11 times that for the nine months up to 30 November (since the first deaths in Europe in late February), which would require a daily death rate over the next year that is 8 times that for the nine months up to 30 November. This does not seem plausible. Furthermore, the cross-country average daily average death rate per 1m over the last nine months (to 30 November) has been 450/270 = 1.7, and 8 times this is therefore 13 per day. By contrast, over the past month (31 October to 30 November), during which the daily death rate in Europe has been substantially higher than for the previous few months, the cross-country average death rate per 1m for these countries has gone from 256 (up to 31 October) to 450 (up to 30 November), which is 7 per day per 1m over November. So, to achieve N = 11 and hence a cross-country average daily rate of 13 over the next year, the cross-country average daily loss in life over the next year would have to be double the daily average over November. Again, this does not seem plausible.Thus, with the benefit of information that has arisen since the first lockdown, the conclusion that mitigation would have been a much better choice than lockdown in March still holds. This conclusion is strengthened by the fact that many of the parameter estimates in equation (5) are from the end of their probability distributions that yield the lowest possible cost per QALY, most particularly the 6,240 extra QALYs lost from mitigation and the $27b GDP loss due to lockdown rather than mitigation. The conclusion is also strengthened by the fact that no allowance has been made for various phenomena that would raise the costs of lockdowns but cannot be quantified: problems arising from the increased unemployment (addiction, crime, domestic violence, mental health problems, and premature death), disruption to the normal operation of the health care system leading to deaths that would not otherwise occur, disruption to the education of the Covid-19 student cohort, and the deprival of liberties that people would otherwise enjoy. 3.5 Evolution in the Views of the Government’s AdvisersIt is interesting to observe the evolution in the views of advisers to the government whose views have been well publicised. At the commencement of the crisis, professors Blakely, Baker and Wilson supported a quantitative cost-benefit analysis of the merits of different possible strategies (Blakely et al, 2020). Consistent with this, they supported the same quantitative approach to health interventions in the pre-covid period; for example, Kvizhinadze et al (2015) estimated the maximum amount that should be spent as a function of a patient’s age on life-saving interventions that returned the patients to average mortality and morbidity for their age group, Boyd et al (2018) estimated the net present value of a border closure in response to a generic pandemic threat, and Boyd et al (2019) estimated the cost per QALY saved by cataract surgery. However, all subsequent published comment on the pandemic from these public health experts (professors Blakely, Baker and Wilson) has been in support of an elimination strategy but without any accompanying cost-benefit analysis, or even the merest mention of QALYs (for example: Baker et al, 2020; Durrheim and Baker, 2020). So, despite promoting the cost-benefit approach to health interventions for many years, and adopting the same approach in the initial stages of the pandemic, these public health experts seem to have abandoned this approach in favour of the goal of minimizing lives lost regardless of the cost. The case of Professor Baker is even stranger. He was a co-author of Blakely et al (2020), which was published on 23 March (and it cannot have been finalized before 16 March because it cited an article with that date). In this article, the authors noted that substantial savings in life were likely from elimination but that there were costs of doing so in the health system and across the economy, that expenditures of up to $2.5b were justified by a QALY value of $45,000, that these were “quick calculations”, and that they “could and should be improved”. However, despite so urging government to improve the cost-benefit analysis so that a sensible decision could be made, Prof Baker was already advocating lockdowns and doing so without any cost-benefit analysis. In particular, he advocated lockdown on 22 March whilst recognizing the severe economic costs of doing so but without any analysis of the trade-off. Furthermore, in an interview granted to a journalist after the lockdown announcement, explaining the evolution in his thinking, he stated that he switched in late February to favouring lockdowns to achieve elimination because the evidence from China was that such a strategy could succeed, but he makes no mention of the costs of doing so or even that cost considerations were important. Consistent with this newfound absolute prioritization of lives over costs, he reports weeping when the lockdown was announced (on 23 March). Turning now to James et al (2020a), in March 2020, they presented predictions of the death tolls in New Zealand from a range of possible control strategies (ibid, Table 2). However, none of the strategies examined by them corresponds to “mitigation” as defined by Ferguson et al (2020) and Blakely et al (2020). Furthermore, James et al did not present any information on, or even recognize the relevance of, costs of adopting the various strategies considered by them, and therefore their goal seemed to be to minimize the number of deaths regardless of the cost. By contrast, in a November 2020 paper with most of the same authors as James et al (2020a), Binny et al (2020, page 19) do recognize that differing strategies have different costs, and that a cost-benefit analysis would be necessary to determine this, but they do not carry this out. In the absence of such analysis, the usefulness of their analysis to policy makers is minimal unless those policy makers are indifferent to costs and seek only to minimise the death rate. Furthermore, as with their earlier paper, none of the strategies examined in their latest paper examines the consequences of a mitigation strategy as defined by Ferguson et al (2020), involving case isolation, quarantining of members of their households, and efforts to shield the elderly. The nearest such strategy examined by them appears to be the adoption of only Level 2. Furthermore, in estimating the death toll from adoption of only Level 2, Binny et al (2020, Table 3) estimate it at 32,000 (and more if the health system is saturated with cases, as it would be at that casualty level). However, their model generating this result has no provision for individuals not yet infected changing their behavior in response to the observable spread of the virus, in the form of reduced social contacts, working from home if possible, increased hand washing, increased use of masks, etc. Implicitly and implausibly, they treat uninfected humans like rabbits, who do not react to a virus spreading through their population because they do not understand how it is transmitted. Equivalently, they treat uninfected human beings like fissile elements in a nuclear chain reaction, which are unable to move so as to avoid the neutrons that facilitate the chain reaction. Consistent with this highly implausible feature of the model, its prediction of at least 32,000 deaths in New Zealand (6,400 per 1m) exceeds the average death rate in European countries up to 30 November (of 450 per 1m) by a multiple of 14, and even exceeds the highest death rate observed in any country in the world up to 30 November (Belgium with 1,418 per 1m) by a multiple of over 4. Interestingly, this (implausible) multiple of 14 would lie above the multiple of 11 referred to in the penultimate paragraph of the previous section, which would justify the March lockdown using the highest possible value per QALY of $146,000.The Auckland LockdownOn 12 August, and consequent upon an outbreak of community cases in Auckland that did not seem capable of being contained with contact tracing and testing, the government locked down the Auckland region for 18 days. Had lockdown not been undertaken, the outbreak may have spread throughout the country, yielding a situation comparable to that on 23 March.This section considers the merits of that decision, based upon information available at the time and retrospectively.I start with information available at the time. In respect of the additional QALYs saved from the Auckland lockdown decision, the 6,240*2*N estimate from the previous section is still the upper bound from continued pursuit of a lockdown policy here, for reasons given there and for four additional reasons. Firstly, the outbreak might have been contained through continued testing and tracing, in which case a mitigation policy might not have led to all of these deaths. Secondly, policy and medical lessons have been learned since the first lockdown that would have reduced deaths if a mitigation policy had been adopted on 12 August. Thirdly, since mass vaccination will bring deaths to an end at some future date T, failure to lockdown in August would then expose New Zealand only to deaths from August 2020 till T whereas the figure of 6,240*2*N is the deaths from failure to lockdown in March 2020 till T, and the former death figure would be less than the latter. Fourthly, the figure of 6,240 above arises from 312 extra deaths per 1m from mitigation, but is based upon information available on 30 November. Based upon information available on 12 August, the figure is considerably lower as noted in footnote 36. The lower bound on the extra QALYs saved from continuation of the lockdown policy is zero. So, the upper bound on the QALYs saved by the lockdown is 6,240*2*N. In respect of the additional GDP loss from the Auckland region lockdown decision, this lasted 18 days (12 August – 30 August in Level 3) whereas the first lockdown lasted 51 days (23 March till 13 May in Levels 3 and 4), and the Auckland region generates 38% of New Zealand’s GDP. In addition, all of the Auckland lockdown was at Level 3 whilst 33 of the 51 days of the first lockdown were at Level 4. Assuming Level 4 curtails output by 60% more than Level 3 (see The Treasury, 2020a, page 7), the first lockdown was then 39% more stringent than the second, on average. So, the Auckland lockdown for only 18 days at Level 3 was equivalent to 10.0% of the first lockdown (18/51*0.38/1.39). Substituted into equation (5), the cost per QALY saved from continued pursuit of the lockdown policy would then be as follows: C= $27b0.10-$1.1b-$0.6bN6,240(2)N (6)For this to be less than the QALY value of $62,000, which would justify lockdown, N would have to be at least 1.2. This is quite likely. However, this conclusion rests upon using estimates for the parameters in equation (6) that are most favourable to a lockdown decision. It also rests upon ignoring various phenomena that would raise the costs of lockdowns but cannot be quantified: problems arising from the increased unemployment (addiction, crime, domestic violence, mental health problems, and premature death), disruption to the normal operation of the health care system leading to deaths that would not otherwise occur, disruption to the education of the Covid-19 student cohort, and the deprival of liberties that people would otherwise enjoy. It also assumes that on 12 August it was reasonable to believe that no further outbreaks would occur elsewhere in the country or again in Auckland that would have led to further lockdowns under the lockdown policy. Taking account of these additional factors, it is not possible to clearly conclude either way on the merits of the Auckland lockdown decision.Repeating this process using information available on 30 November, the result is the same.James et al (2020b) have also examined the Auckland lockdown decision, but focused upon the relative merits of Level 3 versus 4 lockdowns. In their earlier paper concerned with the March lockdown decision, James et al (2020a) did not present any information on, or even recognize the relevance of, the costs of adopting the various strategies considered by them, and therefore their goal seemed to be to minimize the number of deaths regardless of the cost. By contrast, James et al (2020b) estimate the economic costs of Level 3 and 4 lockdowns, and in particular the GDP losses that would have been incurred under Level 3 and 4 lockdowns if each were sustained for the period necessary to raise the probability of eliminating the outbreak to 50%. They find a modest such advantage to Level 4, because the expected time in lockdown to reach their epidemiological target is shorter in Level 4, which more than compensates for the higher costs per day. However, and despite referring to the use of the QALY approach in other papers and even appearing to recommend such an approach to the Auckland decision (bid, page 10), they do not adopt it themselves nor offer a reason for failing to do so. Furthermore, their GDP losses are merely for the days in which the lockdown operates, and therefore they implicitly assume that removal of restrictions leads to the immediate reversion of GDP to the path it would otherwise have followed. However, as shown in Table 1 and Table 4 above, it is expected to take several years to fully recover from a lockdown and there is no reason to suppose the entire such costs are the same multiple of the costs during the lockdown for both Level 3 and 4. In addition, they attribute all GDP losses to the lockdown, whereas a substantial fraction would be due to actions taken by individuals in response to an outbreak, and the fraction will likely differ across Levels 3 and 4 lockdowns.In conclusion, it is not possible to clearly conclude either way on the merits of the Auckland lockdown decision, based upon information available both at the time and subsequently.Looking Forward on 30 NovemberI now consider the appropriate course of action looking forwards at the present time (30 November). Deaths and GDP losses to date are sunk, and therefore should be ignored. Since the Auckland lockdown there have been a number of cases in which a person outside the border quarantine zone has tested positive (“community cases”). These cases have been dealt with so far by quarantining of the infected individuals and any of their contacts also found to be infected. However, at any time, such an approach might prove to be insufficient to contain an outbreak, and therefore the question of lockdown would again arise (as it did with the August Auckland lockdown decision). Any such lockdowns may be only regional rather than national. Furthermore, since viruses generally evolve over time, the death rate arising from any future outbreaks without a lockdown may be more or less severe than indicated by currently available data. I therefore consider both issues.Let P denote the proportion of the national GDP generated by the region subject to lockdown, so P = 1 means lockdown of the entire country. Following equation (5), the $27b would be proportional to P. So, the expected cost per QALY saved from a lockdown rather than a mitigation policy would then be as follows: C=$27bP-$1.1b-N$0.6b6,240(2)N (7)This presumes that the incremental deaths resulting from failure to lockdown at some future point would be 6,240*2*N. However, as discussed in the previous section, since mass vaccination will bring deaths to an end at some future date T, failure to lockdown at some future point T1 prior to T would then expose New Zealand only to deaths from T1 till T whereas the figure of 6,240*2*N is the deaths from failure to lockdown in March 2020 till T, and the former death figure would then be less than the latter. So, failure to lockdown at some future point T1 will yield incremental deaths of 6,240*2*N*Q, where Q < 1 reflects the reduction in deaths resulting from the fact that the exposure period T1 till T is shorter than that from March 2020 till T. The expected cost per QALY saved from a lockdown rather than a mitigation policy would then be as follows: C=$27bP-$1.1b-NQ$0.6b6,240(2)NQ For this to be less than the threshold figure of $62,000, which would support lockdown, then NQ≥P$27b-$1.1b$1.37b (8) So, if P = 1, NQ would have to be at least 19, and therefore N even larger than this. As argued in the previous section, even N = 19 is implausibly large. However, if P = .10 as with the Auckland lockdown, NQ would then only have to be at least 1.2, which is more plausible. As P goes to zero, with quarantining of a single person being the extreme case, the lower limit on NQ falls below zero, in which case lockdown is definitely optimal.This analysis assumes that, if lockdown is pursued, only one future lockdown would be necessary in New Zealand. This cannot be guaranteed, and two lockdowns have already occured. So, the GDP losses from a lockdown policy may exceed $27b and cannot be less. In so far as they are more, for a given value of P, even higher values of N would be required to justify a lockdown policy. For example, if there is a 50% chance of a second future lockdown involving the same value for P, and in that event there is a 50% chance of a third future lockdown and so on, the last inequality becomes NQ≥P$27b2-$1.1b$1.37b (9)So, if P = .10 consistent with the Auckland lockdown, NQ would have to be at least 3.1 to justify lockdown, and N even larger than this. This value for NQ is much larger than before and therefore much less likely to be true. Finally, as time passes, the point in time T1 at which a lockdown might be undertaken in response to an outbreak moves closer to the mass vaccination point T, and therefore the ratio (T - T1)/(T – March 2020) falls, which means that Q falls. So, satisfying (8) or (9) becomes even harder, and therefore would only be warranted for smaller and smaller values of P. For example, suppose that (8) prevails (one future lockdown at most), N = 2, mass vaccination will occur in one year from now, and Q is the ratio of the time to mass vaccination (currently 1 year) to the interval from March 2020 till mass vaccination (1.75 yrs) so that Q = 1/1.75 = 0.57 currently. Inequality (8) will then be satisfied for P ≤ 0.10. However, in four months’ time, Q = 0.67/1.75 = 0.38 in which case inequality (8) will only then be satisfied for P ≤ 0.08. So, as we move closer to the mass vaccination point in time, the proportion of the national GDP generated by the region subject to a lockdown that is justified declines towards zero.As before, these conclusions rest upon using estimates for the parameters in equation (5) that are most favourable to a lockdown decision. These conclusions also rest upon ignoring various phenomena that would raise the costs of lockdowns but cannot be quantified: problems arising from the increased unemployment (addiction, crime, domestic violence, mental health problems, and premature death), disruption to the normal operation of the health care system leading to deaths that would not otherwise occur, disruption to the education of the Covid-19 student cohort, and the deprival of liberties that people would otherwise enjoy. Allowance for these factors means that, for a given value of P, the minimum value of N consistent with a lockdown policy being optimal rises, thereby making lockdown less likely to be justified.In conclusion, in considering the possibility of future outbreaks that could not be contained with only quarantining of cases, lockdowns might be justified if the proportion of the national GDP generated by the region subject to lockdown were sufficiently small (with Auckland currently being the upper limit) and the probability of conditions giving rise to a second future lockdown is not high. As we move closer to the mass vaccination point in time, this proportion goes to zero.ConclusionsWorld-wide, many governments have implemented substantial curtailments of normal economic activity in order to reduce the expected death toll from Covid-19. This paper considered the merits of the New Zealand government adopting a lockdown rather than a milder mitigation policy, with the actions of individuals and other governments taken as given. Based upon the information available at the time of the first lockdown in March, the cost per QALY saved from doing so was well in excess of generally used values for a QALY. Consideration of alternative parameter values and recognition of factors omitted from the analysis would not likely reverse this conclusion. The lockdown policy was therefore inconsistent with generally used values for a QALY. This paper then considered the implications of information that subsequently became available, most particularly the death rate evidence from other countries. This information reinforced the conclusion that the March lockdown was unwarranted. The paper then considered the merits of the subsequent Auckland lockdown. It is not possible to clearly conclude either way on the merits of this decision, based upon information available both at the time and subsequently.Finally, in considering the possibility of future outbreaks that could not be contained with only quarantining of cases, lockdowns might be justified if the proportion of the national GDP generated by the region subject to lockdown were sufficiently small (with Auckland currently being the upper limit) and the probability of conditions giving rise to a second future lockdown is not high. As we move closer to the mass vaccination point in time, this proportion goes to zero.APPENDIX 1This Appendix further examines data on death rates across countries that did and did not lockdown. Given the actual or probable unreliability of data from many countries, it is desirable to limit the analysis to countries for which the data is likely to be very reliable. It is also desirable to eliminate countries with federal systems, in which restrictions would vary by state or province. This leaves European countries and the East Asian democracies (Japan, South Korea, Taiwan, Singapore, plus Hong Kong). Hale et al (2020) have constructed a set of indexes, which assign a daily score to each country for the severity of their restrictions imposed by government, ranging from 0 to 100 and taking account of different types of restrictions. I use their Stringency Index, which takes account of 8 different types of government restrictions, and use the maximum rating for each country assigned during the crisis. Death rates per 1m of population are drawn from the previously mentioned source. All five East Asian countries have very low death rates regardless of the severity of restrictions, and the likely reasons (a culture of mask wearing, not shaking hands, compliance with government directives, extensive contact tracing and testing, etc) are or were not applicable to the same degree in New Zealand. So, I use only the European countries, of which there are 33. They are similar (on average) to New Zealand in ethnicity, cultural norms, demographics, GDP per capita, and the quality of their health care systems. Unlike Gibson (2020), I use the maximum index value rather than the average. This avoids the difficulties in choosing a starting date for the averaging. Gibson uses the same date for all countries (7 January) but the first cases appeared in countries at different times, which will produce lower average index values in countries that were hit later rather even if their true policy response was equally stringent. Use of the maximum value also avoids the problem that the average over the entire crisis period is affected by the post-peak death rate reductions in the index, which are likely to have been affected by the death rate (up to that point) as well as having contributed to it (beyond that point). This yields a reverse causality problem. An extreme case of this problem would arise if one country adopted its maximum stringency value of 100 on the first day, retained it for six weeks and then removed all restrictions because eradication had been achieved, whilst a second country imposed a stringency of 50 throughout the 12 weeks of the analysis. Both would have an average Stringency of 50, but would embody entirely different policies.Death rates are likely to be affected by many variables other than government restrictions, and it is desirable to include them. I consider population density (higher values increase the transmission rate of the virus), the date of the first death (in days after the first recorded death on 15 February in France), because later dates provide more time for people, doctors and their governments to learn from others and adjust their behavior, population (higher values provide a higher pool of virus targets before national borders constrict the movement of people and therefore the transmission of the virus),GDP per capita (as a proxy for the quality of the health care system), the population proportion over 65 (higher values imply a larger proportion of the population in the high risk group), the average household size (higher values increase the pool of virus targets before household borders restrict interactions and therefore the transmission of the virus), and the number of nursing and elderly home beds per 100,000 of population (because higher values implies a higher concentration in the high-risk group, which increases or lowers the death rate depending upon the effectiveness of the quarantine procedures).Flu intensity in the last two flu seasons. The first two variables are statistically significant, and substantially raise the adjusted R2, whilst the last five variables (added and tested separately) are not statistically significant and their inclusion each lowers the adjusted R2. I therefore retain only the first two variables. The data are shown in Appendix 2. Regressing the death rate per 1m (D) up to 22 August on the maximum Stringency Index value (S), the population density (PD, in millions per 1,000 square miles), and date of first death (FD, in days from 15 February) yields the following result: D=569.3-1.65S+299.7PD-11.9FD (10)The R2 is a respectable 0.49, all coefficients on the regressors have the expected signs, and the p values are .05, .59, .03 and .002 respectively. In particular, the death rate falls as the stringency of the restrictions rises. However, unlike the other three coefficients in the model, the coefficient on S is not strongly statistically significant (p = 0.59), so the hypothesis that D is unrelated to S cannot be rejected. Furthermore, if Sweden is deleted from the regression model (1/33 data points), the coefficient on S becomes positive. So, the evidence for government restrictions reducing the death rate is minimal. This may seem counterintuitive, but the explanation may lie in the adjective. Even without government restrictions, people will take actions to lower their risks in a pandemic and the incremental effect of government actions may then be too little to be statistically significant. This is consistent with the economic evidence that most of New Zealand’s expected GDP losses from the pandemic are not attributable to the actions of the New Zealand government.The epidemiological impact of all three independent variables on D is substantial relative to the observed variation in D (which ranges from 6 to 850); going from the minimum to the maximum value for S (0.54 to 1.0) is expected to lower D by 76 per 1m, going from the minimum to the maximum value for PD (0.008 to 1.06) is expected to raise D by 315 per 1m, and going from the minimum to the maximum value for FD (1 to 46) is expected to lower D by 536 per 1m. Interestingly, using this model, the predicted death rate (to date) for New Zealand (with its regressor values of S = 96, PD = 5.0/103 and FD = 42) is negative, so New Zealand (with an actual death rate of 4) underperformed relative to its regressor values.It is fundamental to equation (10) that causality runs from the regressors to the death rate. However, it is possible that the causality also runs from D to S because some governments chose their S value at the commencement of the crisis based upon their predictions of D under both low and high S scenarios, and/orsome governments chose their S values based upon their observation of their country’s death rate in the early stages of the crisis. If either of these is true, the estimated coefficient on S in equation (8) may be biased. The traditional method of dealing with this is to use an “instrumental variable”, but no good candidates are apparent. I therefore enquire into the extent of these problems. In respect of the first possible problem, I will focus upon the death rates under mitigation (S = 50) and extreme lockdown (S = 100). Suppose that there are two types of countries (A and B) whose governments held the views shown in Table 6 (at the commencement of the crisis) about expected death rates under mitigation and extreme lockdown.Table 6: Expected Death Rates under Various Scenarios___________________________________________________________________________S = 50S = 100___________________________________________________________________________Country A2,040170Country B250< 170___________________________________________________________________________ The data for type A countries with S = 100 and type B countries with S = 50 is taken from equation (10) with the other regressors fixed at their average values (PD = 0.313 and FD = 28). The death rate for type A countries under mitigation (2,040) is 12 times that of their death rate under extreme lockdown, with the multiple reflecting the predictions of Ferguson et al (2020) for the UK, i.e., 20,000 deaths under suppression (which is 300 per 1m of the UK’s population of 68m) and 250,000 deaths under mitigation (which is 3,700 per 1m of the UK’s population). Type A countries chose S = 100 because the expected death rate is unacceptably high with S = 50. Type B countries have much lower expected death rates than type A countries under both mitigation and extreme lockdown, and chose S = 50 because the expected death rate in that scenario is acceptable. If the governments’ predictions are on average accurate, then the outcomes would be as follows:Type A countries: S = 100 and D = 170 on averageType B countries: S = 50 and D = 250 on averageRegressing D on S would then yield a coefficient on S of -1.6, consistent with the regression result shown in equation (10). However, the real coefficients on S are much lower, most particularly -37.0 for type A countries. Thus, the regression coefficient on S would be biased upwards.It is plausible that some governments believed that their death rate under mitigation would be both large and vastly in excess of their death rate under extreme lockdown, as shown in Table 5, and acted accordingly in accordance with the predictions of experts like Ferguson et al (2020). It is also plausible that other governments believed that their death rates under mitigation would be much lower, as shown in Table 6, and acted accordingly in accordance with contrary expert opinions. However, both types of governments’ beliefs would need to be (on average) correct in order to be compatible with the coefficient on S in equation (8) of -1.6. Thus, there would have to be features of these two types of countries that would justify the markedly higher death rate in type A countries than in type B countries under mitigation (times 8 in Table 6), and governments would have to have been capable of recognizing these at the commencement of the crisis. Experts’ predictions, such as those of Ferguson et al (2020), would not have helped. For example, Ferguson et al (2020) used Chinese data to generate predictions for only the UK and US, which differed only slightly (3,700 per 1m for the UK and 3,500 for the US) due to demographics and population density (ibid, pp. 6-7 and 16). Furthermore, his numerous critics believed his death rates for the UK and US under mitigation were too high rather than that they were correct for those countries but far too high for others. Furthermore, if by some other means governments believed that their death rates under mitigation would markedly differ due to some variable other than the regressors used in equation (10) or those tested and rejected by me, and their beliefs were correct, they would have to have been aware in advance of a variable that I have not been able to locate even with the advantage of subsequently obtaining and testing the data that has become available since the commencement of the crisis. These conditions are not plausible, and this implies that the estimated coefficient on S of -1.65 is not materially biased for reasons of this type.The second potential problem with equation (10) is that some governments may have chosen their S values in light of their observation of their country’s death rate in the early stages of the crisis, believing it would predict the final death rate. To illustrate this, suppose there are two types of countries, with average death rates under mitigation and extreme lockdown thus:Table 7: Average Death Rates under Various Scenarios___________________________________________________________________________S = 50S = 100___________________________________________________________________________Country A600170Country B250< 170___________________________________________________________________________ At the commencement of the crisis, it is unknown which category each country lies in but it is revealed by the death rates in the early stages of the crisis. So, upon observing their early stage death rates, the governments of type A countries then understood that they were of that type and chose extreme lockdown, yielding S = 100 and an average death rate of D = 170. At the same point, the governments of type B countries then understood that they were of that type and chose mitigation, yielding S = 50 and an average death rate of D = 250. Regressing D on S would then yield a coefficient on S of -1.6, consistent with the regression result shown in equation (10). However, the real coefficients on S are much lower, most particularly -8.6 for type A countries. Thus, the regression coefficient on S would be biased up.This scenario can be tested as follows. For each country, I regress its Stringency value ten days after its first reported death (S10) on its death rate up to that point (D10), to assess whether D10 can explain S10. I repeat the process for 20 and 30 days after each country’s first death. I also test whether any of these three early stage death rates can explain the maximum S value chosen by governments (Sm). These regressions yield the results shown in the first six columns of Table 8 below. Only two of these regressions even yield a positive coefficient on early death rate (consistent with the scenario in Table 7) and none of them yields a statistically significant coefficient on it. So, the hypothesis that early stage death rates did not affect governments’ choice of S cannot be rejected. This is surprising because the death rates up to day 20, and even more so for up to day 30, are good predictors of the final death rate (D), as shown in the last three columns of Table 8. So, at least from day 20, the death rate data up to that point could have been used to set the S value at that point or the maximum S value but governments did not seem to have done so. Table 8: Stringency Levels and Death Rates___________________________________________________________________________Dep Var (DV)S10S20S30SmSmSmDDDIndep Var (IV)D10D20D30D10D20D30D10D20D30___________________________________________________________________________Mean DV value687779818181198198198Mean IV value317503175031750Coeff on DV1.9-.07.002-.57-.10-.027.87.42.8P Value for DV.09.59.96.37.37.52.5300Adjusted R2.06-.02-.03-.01-.01-.02-.02.40.66___________________________________________________________________________ This raises the interesting question of what does then explain the maximum S values. Regressing this variable on the variables used in or tested for inclusion in equation (10), being population density, date of first death, proportion of population over 65, beds per 100,000 people, GDP per capita, and household size, yielded no statistically significant coefficients. However, ranking the maximum S values from highest to lowest (see Appendix 2) reveals that the four countries arising from the breakup of Yugoslavia occupy four of the ‘top’ six slots (with an average S value of 95) and the five Scandinavian countries occupy four of the ‘bottom’ five slots (and have an average S value of 68). This suggests that the S choice was in part driven by mimicry of neighbouring countries. Consistent with this, Sebhatu et al (2020) finds that the speed with which restrictions were adopted by the OECD members was influenced by the behavior of nearby countries.A further criticism of equation (10) is that many other independent variables have not been included, because of data limitations, and inclusion of them could materially alter the coefficient on S. Consistent with this, the adjusted R2 for equation (10) is 0.44. Examples of these omitted variables are the extent of multi-generational households, which would facilitate the transmission of the virus to the high risk elderly group, and the extent of quarantining of nursing homes, which would reduce the death rate amongst the high risk group. However, starting with the version of equation (10) with only S as the regressor and all countries included, and then progressively adding new regressors, the change in the estimate of S is not very substantial, with values ranging from xxx to xxx. All of this suggests that, if further regressors could be added, the coefficient on S would not be materially less than -1.65. With this value, the expected increase in New Zealand’s death toll from moving from its lockdown policy (S = 96) to a mitigation policy alike that of the least restrictive European country (Iceland with S = 54) would be to raise the death rate by -1.65*(54 – 96) = 69 per 1m of population, which would yield an increase in New Zealand’s deaths of 69*5 = 345. Mindful that the coefficient of -1.65 is merely an estimate, and that there may have been some upward bias for the reasons described earlier, the most conservative approach would be to instead adopt the lower 95% confidence limit bound on the S coefficient, which is -7.92. In this case, reducing New Zealand’s S value from 96 to 54 would raise the death rate by -7.92*(54 – 96) = 333 per 1m of population, which would yield an increase in New Zealand’s deaths of 333*5 = 1,665. This analysis uses deaths up to 22 August. Since then (to 30 November), deaths have risen significantly, from an average of 195 per 1m to 450 per 1m. Repeating the analysis with deaths up to 30 November, the result is as follows: D=376.3+3.43S+462.0PD-12.6FD (11)So, relative to equation (10), the coefficient on S rises from -1.65 to 3.43 and the lower bound on the 95% confidence interval rises from -7.92 to -6.41. Using the lower bound on the 95% CI, the expected increase in New Zealand’s death toll from moving from its lockdown policy (S = 96) to a mitigation policy alike Iceland’s (S = 54) would be to raise its death rate by -6.41*(54 – 96) = 269 per 1m of population, which would yield an increase in New Zealand’s deaths of 269*5 = 1,345. This figure of 1,345 is smaller than from data up to 22 August (1,665). Given the trend since 22 August, this figure of 1,345 is likely to become even smaller as the pandemic unfolds.This analysis also uses data from European countries, because the quality of the data is judged to be sufficient, and conservatively excludes East Asian democracies with very low death rates regardless of government policy because cultural norms may differ significantly from New Zealand. As a robustness test, I examine the next best source of data, which appears to be that from the Americas, but with the exclusion of the US and Canada (because they are federal systems with variation in policy by state) and also Cuba, Nicaragua and Venezuela (because authoritarian regimes are likely to deliberately understate deaths). I also exclude countries with less than 50,000 people because death rates expressed per 1m of population (as the data source does) are only then expressible in multiples of 20 or more. Across the countries for which both Hale et al (2020) provides the Stringency data and the website provides death rate data, there are 28 countries: Brazil, Argentina, Colombia, Mexico, Peru, Chile, Ecuador, Bolivia, Panama, Dominican Republic, Costa Rica, Guatemala, Honduras, Paraguay, El Salvador, French Guyana, Jamaica, Haiti, Trinidad and Tobago, Suriname, Aruba, Guyana, Belize, Uruguay, Cayman Islands, Barbados, Bermuda, and Dominica. Unlike the European data, population density and date of first death are not statistically significant, but the following two regressors are statistically significant: population (higher values provide a higher pool of virus targets before national borders constrict the movement of people and therefore the transmission of the virus),having no land borders with other countries (water barriers rather than land borders better restrict the flow of people and hence the virus into a country).The coefficients on these latter two variables each have the expected sign (positive and negative respectively). If the maximum Stringency index is added, it is not statistically significant and the estimated coefficient on it is positive rather than the expected negative. With P denoting population in millions and I denoting no land borders (1 if so and 0 otherwise), the model inclusive of the maximum Stringency Index (S) is thus: D=-495+9.64S+3.45P-256I (12)The R2 is a respectable 0.47, and the p values on the four coefficients are .37, .11, .005 and .035 respectively. If the European and Americas data are pooled, P, I and date of first death (FD) are statistically significant at the 10% level, with all coefficients having the expected signs. Addition of S yields a coefficient that is not statistically significant, and with the wrong sign. The resulting model is thus: D=232+4.19S+2.96P-214I-5.8FD (13)The R2 is a respectable 0.37, and the p values on the five coefficients are .44, .24, .006, .052, and .040 respectively. Across these three models (11), (12) and (13), equation (11) then yields the lowest coefficient on S, and use of it is therefore conservative. This additional analysis also amplifies the point that New Zealand was in a remarkably favourable position, even without resort to lockdowns, because it has favourable values for all of the variables found to reduce the Covid-19 death rate: low population, no land borders, low population density, and first death well after the first death in Europe.A further feature of the European data is noteworthy. The increase in the death rate per 1m from 22 August to 30 November for European countries has varied significantly across countries, from a mere 14% more in the case of Sweden (576 to 660) to a multiple of 24 in the case of Slovakia (6 to 149). Cross-sectionally regressing the change in proportional terms on the 22 August death rate (D) yields a negative and statistically significant coefficient on the 22 August death rate (p = .015):?DD=5.80-.010DSo, a country with an August 22 death rate of 100 per 1m would be expected to experience a fivefold increase in that rate by 30 November whilst one with an August 22 death rate of 550 per 1m would expect to experience only a 30% increase. A plausible explanation for this is that a country’s death rate up to a point in time reflects inter alia the collective actions of individuals and government (“interventions”) up till that time, such interventions cannot be sustained at high levels for very long, and therefore countries with low death rates on 22 August (tending to be those with stronger interventions up to that time) subsequently experienced higher growth rates in their death rates as their interventions were inevitably scaled back. This suggests that interventions affect only the timing of deaths rather than the number. This conclusion is not inconsistent with the conclusion that lockdowns do not have a statistically significant effect on the death rate because lockdowns (which are a government action) are only part of the total interventions by individuals and governments.The application of the coefficient on S in equations (11), (12) and (13) to New Zealand presumes that the same value applies to all countries. However, New Zealand is in an unusual position, in that the first lockdown seemed to have achieved eradication of community cases, and likewise for the second lockdown. By contrast, most of the European countries that implemented lockdowns never seemed to have achieved eradication of community cases, with the result that their death rates were much higher than New Zealand. This implies that the estimate of S derived from these countries would underestimate the value for New Zealand. The extreme underestimation scenario here involves lockdowns by European countries having no effect upon their death rates, consistent with the evidence presented above. In this hypothetical situation, a different approach would be warranted to estimate the difference in lives lost under lockdown versus mitigation in New Zealand. In respect of lockdown, the death rate to date is 5 per 1m, and this represents a lower bound on the death rate from lockdown in New Zealand (because more deaths may be to come). In respect of mitigation in New Zealand, and on the basis that lockdowns were ineffective in Europe, an estimate of the New Zealand death rate under mitigation would be the average European death rate, corrected for differences in variables that are statistically significant in explaining the death rate. For Europe, these are population density and date of first death, and the resulting model using death rate data to 30 November is D=663+473PD-11.8FD (14)Substitution of New Zealand’s values for the regressors, of PD = .0485 and FD = 42, yields an estimated death rate under mitigation of 144 per 1m. A very conservative estimate would be the upper bound on the 95% CI for this prediction, and this is 317 per 1m. Deduction of the actual death rate under lockdown of 5 per 1m yields a saving in lives from lockdown over mitigation of 312 per 1m. This is larger than the figure of 269 per 1m estimated from equation (11) above. Unlike the latter figure of 269, the figure of 317 is also growing as more deaths ensue in Europe. For both reasons, this approach is preferred so as to be conservative. Since the figure of 317 is growing approximately in proportion to the cross-country average death rate in Europe, the estimated reduction in the death rate for New Zealand under lockdown rather than mitigation is 317N – 5, where N is the final cross-country average death rate in Europe as a proportion of the current (30 November) rate of 450. For example, if the final death rate is 1000 per 1m, the estimated death rate for New Zealand under mitigation would be 317(1,000/450) - 5 = 700 per 1m. This figure is likely to be far too high because the 317 figure is the upper bound on the 95% CI and no further deaths under the current lockdown policy are allowed for.APPENDIX 2Table 9: Covid-19 Death Rates up to 22/8 and Explanatory Factors for European Countries___________________________________________________________________________StringencyFirst DeathDensityDeath Rate___________________________________________________________________________Albania90250.2685Austria81260.2881Belgium81250.98850Bosnia93350.17160Bulgaria73250.1677Switzerland73190.55231Croatia96330.1941Czech Rep82360.3538Cyprus94350.3317Germany77230.61111Denmark72280.35107Spain85170.24617Estonia78390.0747Finland68350.0460France8810.31467UK80190.73609Greece84260.2023Hungary77290.2763Ireland91250.18359Iceland54350.00829Italy9470.52586Lithuania87340.1131Luxembourg80270.06198Latvia69480.0818Netherlands80201.06361Norway80260.0449Poland83260.3151Portugal88300.29176Romania87360.21166Serbia100340.2679Slovakia87460.296Slovenia90280.2763Sweden65250.06575___________________________________________________________________________ReferencesAguas, R., and Corder, R., 2020. “Herd Immunity Thresholds for SARS-CoV-2 Estimated from Unfolding Epidemics”, , A., Hansen, E, Johannesen, N., and Sheridan, A., 2020. “Pandemic, Shutdown, and Consumer Spending: Lessons from Scandinavian Policy Responses to Covid-19”, , D., and Hamilton, F., 2020. “Patient Outcomes after Hospitalisation with Covid-19 and Implications for Follow-Up: Results from a Prospective UK Cohort”, , M., Kvalsvig, A., Verrall, A, Telfer-Barnard, L., and Wilson, N., 2020. “New Zealand’s Elimination Strategy for the Covid-19 Pandemic and what is Required to make it Work”, New Zealand Medical Journal, vol. 133, , A., Clegg, J., Thuresson, P., Lloyd, A., and MacEwan, P., 2020. “Review of Utility Values for Economic Modelling in Type 2 Diabetes”, Value in Health, vol. 17, pp. 462-470, , D., and Comas-Herrara, A., and Henderson, D., 2020. “Covid-19 Mortality and Long-Term Care: A UK Comparison, International Long-Term Care Policy Network, HYPERLINK "" , R., Baker, M., Hendy, S., James, A., Lustig, A., Plank, M., Ridings, K., and Steyn, N., 2020. “Early Intervention is the Key to Success in Covid-19 Control”, working paper, , T., Baker, M., and N. 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